Direct linear transformation (DLT) is an algorithm which solves a set of variables from a set of similarity relations:

${\displaystyle \mathbf {x} _{k}\propto \mathbf {A} \,\mathbf {y} _{k}}$   for ${\displaystyle \,k=1,\ldots ,N}$

where ${\displaystyle \mathbf {x} _{k}}$ and ${\displaystyle \mathbf {y} _{k}}$ are known vectors, ${\displaystyle \,\propto }$ denotes equality up to an unknown scalar multiplication, and ${\displaystyle \mathbf {A} }$ is a matrix (or linear transformation) which contains the unknowns to be solved.

This type of relation appears frequently in projective geometry. Practical examples include the relation between 3D points in a scene and their projection onto the image plane of a pinhole camera,^[1] and homographies.

Introduction

An ordinary system of linear equations

${\displaystyle \mathbf {x} _{k}=\mathbf {A} \,\mathbf {y} _{k}}$   for ${\displaystyle \,k=1,\ldots ,N}$

can be solved, for example, by rewriting it as a matrix equation ${\displaystyle \mathbf {X} =\mathbf {A} \,\mathbf {Y} }$ where matrices ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ contain the vectors ${\displaystyle \mathbf {x} _{k}}$ and ${\displaystyle \mathbf {y} _{k}}$ in their respective columns. Given that there exists a unique solution, it is given by

${\displaystyle \mathbf {A} =\mathbf {X} \,\mathbf {Y} ^{T}\,(\mathbf {Y} \,\mathbf {Y} ^{T})^{-1}.}$

Solutions can also be described in the case that the equations are over or under determined.

What makes the direct linear transformation problem distinct from the above standard case is the fact that the left and right sides of the defining equation can differ by an unknown multiplicative factor which is dependent on k. As a consequence, ${\displaystyle \mathbf {A} }$ cannot be computed as in the standard case. Instead, the similarity relations are rewritten as proper linear homogeneous equations which then can be solved by a standard method. The combination of rewriting the similarity equations as homogeneous linear equations and solving them by standard methods is referred to as a direct linear transformation algorithm or DLT algorithm. DLT is attributed to Ivan Sutherland. ^[2]

Example

Suppose that ${\displaystyle k\in \{1,...,N\}}$. Let ${\displaystyle \mathbf {x} _{k}=(x_{1k},x_{2k})\in \mathbb {R} ^{2}}$ and ${\displaystyle \mathbf {y} _{k}=(y_{1k},y_{2k},y_{3k})\in \mathbb {R} ^{3}}$ be two known vectors, and we want to find the ${\displaystyle 2\times 3}$ matrix ${\displaystyle \mathbf {A} }$ such that

${\displaystyle \alpha _{k}\,\mathbf {x} _{k}=\mathbf {A} \,\mathbf {y} _{k}}$

where ${\displaystyle \alpha _{k}\neq 0}$ is the unknown scalar factor related to equation k.

To get rid of the unknown scalars and obtain homogeneous equations, define the anti-symmetric matrix

${\displaystyle \mathbf {H} ={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}$

and multiply both sides of the equation with ${\displaystyle \mathbf {x} _{k}^{T}\,\mathbf {H} }$ from the left

${\displaystyle {\begin{aligned}(\mathbf {x} _{k}^{T}\,\mathbf {H} )\,\alpha _{k}\,\mathbf {x} _{k}&=(\mathbf {x} _{k}^{T}\,\mathbf {H} )\,\mathbf {A} \,\mathbf {y} _{k}\\\alpha _{k}\,\mathbf {x} _{k}^{T}\,\mathbf {H} \,\mathbf {x} _{k}&=\mathbf {x} _{k}^{T}\,\mathbf {H} \,\mathbf {A} \,\mathbf {y} _{k}\end{aligned}}}$

Since ${\displaystyle \mathbf {x} _{k}^{T}\,\mathbf {H} \,\mathbf {x} _{k}=0,}$ the following homogeneous equations, which no longer contain the unknown scalars, are at hand

${\displaystyle \mathbf {x} _{k}^{T}\,\mathbf {H} \,\mathbf {A} \,\mathbf {y} _{k}=0}$

In order to solve ${\displaystyle \mathbf {A} }$ from this set of equations, consider the elements of the vectors ${\displaystyle \mathbf {x} _{k}}$ and ${\displaystyle \mathbf {y} _{k}}$ and matrix ${\displaystyle \mathbf {A} }$:

${\displaystyle \mathbf {x} _{k}={\begin{pmatrix}x_{1k}\\x_{2k}\end{pmatrix}}}$,   ${\displaystyle \mathbf {y} _{k}={\begin{pmatrix}y_{1k}\\y_{2k}\\y_{3k}\end{pmatrix}}}$,   and   ${\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{pmatrix}}}$

and the above homogeneous equation becomes

${\displaystyle 0=a_{11}\,x_{2k}\,y_{1k}-a_{21}\,x_{1k}\,y_{1k}+a_{12}\,x_{2k}\,y_{2k}-a_{22}\,x_{1k}\,y_{2k}+a_{13}\,x_{2k}\,y_{3k}-a_{23}\,x_{1k}\,y_{3k}}$   for ${\displaystyle \,k=1,\ldots ,N.}$

This can also be written in the matrix form:

${\displaystyle 0=\mathbf {b} _{k}^{T}\,\mathbf {a} }$   for ${\displaystyle \,k=1,\ldots ,N}$

where ${\displaystyle \mathbf {b} _{k}}$ and ${\displaystyle \mathbf {a} }$ both are 6-dimensional vectors defined as

${\displaystyle \mathbf {b} _{k}={\begin{pmatrix}x_{2k}\,y_{1k}\\-x_{1k}\,y_{1k}\\x_{2k}\,y_{2k}\\-x_{1k}\,y_{2k}\\x_{2k}\,y_{3k}\\-x_{1k}\,y_{3k}\end{pmatrix}}}$   and   ${\displaystyle \mathbf {a} ={\begin{pmatrix}a_{11}\\a_{21}\\a_{12}\\a_{22}\\a_{13}\\a_{23}\end{pmatrix}}.}$

So far, we have 1 equation and 6 unknowns. A set of homogeneous equations can be written in the matrix form

${\displaystyle \mathbf {0} =\mathbf {B} \,\mathbf {a} }$

where ${\displaystyle \mathbf {B} }$ is a ${\displaystyle N\times 6}$ matrix which holds the known vectors ${\displaystyle \mathbf {b} _{k}}$ in its rows. The unknown ${\displaystyle \mathbf {a} }$ can be determined, for example, by a singular value decomposition of ${\displaystyle \mathbf {B} }$; ${\displaystyle \mathbf {a} }$ is a right singular vector of ${\displaystyle \mathbf {B} }$ corresponding to a singular value that equals zero. Once ${\displaystyle \mathbf {a} }$ has been determined, the elements of matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): \mathbf {A} can rearranged from vector ${\displaystyle \mathbf {a} }$. Notice that the scaling of ${\displaystyle \mathbf {a} }$ or ${\displaystyle \mathbf {A} }$ is not important (except that it must be non-zero) since the defining equations already allow for unknown scaling.

In practice the vectors ${\displaystyle \mathbf {x} _{k}}$ and ${\displaystyle \mathbf {y} _{k}}$ may contain noise which means that the similarity equations are only approximately valid. As a consequence, there may not be a vector ${\displaystyle \mathbf {a} }$ which solves the homogeneous equation ${\displaystyle \mathbf {0} =\mathbf {B} \,\mathbf {a} }$ exactly. In these cases, a total least squares solution can be used by choosing ${\displaystyle \mathbf {a} }$ as a right singular vector corresponding to the smallest singular value of ${\displaystyle \mathbf {B} .}$

More general cases

The above example has ${\displaystyle \mathbf {x} _{k}\in \mathbb {R} ^{2}}$ and ${\displaystyle \mathbf {y} _{k}\in \mathbb {R} ^{3}}$, but the general strategy for rewriting the similarity relations into homogeneous linear equations can be generalized to arbitrary dimensions for both ${\displaystyle \mathbf {x} _{k}}$ and ${\displaystyle \mathbf {y} _{k}.}$

If ${\displaystyle \mathbf {x} _{k}\in \mathbb {R} ^{2}}$ and ${\displaystyle \mathbf {y} _{k}\in \mathbb {R} ^{q}}$ the previous expressions can still lead to an equation

${\displaystyle 0=\mathbf {x} _{k}^{T}\,\mathbf {H} \,\mathbf {A} \,\mathbf {y} _{k}}$   for   ${\displaystyle \,k=1,\ldots ,N}$

where ${\displaystyle \mathbf {A} }$ now is ${\displaystyle 2\times q.}$ Each k provides one equation in the ${\displaystyle 2q}$ unknown elements of ${\displaystyle \mathbf {A} }$ and together these equations can be written ${\displaystyle \mathbf {B} \,\mathbf {a} =\mathbf {0} }$ for the known ${\displaystyle N\times 2\,q}$ matrix ${\displaystyle \mathbf {B} }$ and unknown 2q-dimensional vector ${\displaystyle \mathbf {a} .}$ This vector can be found in a similar way as before.

In the most general case ${\displaystyle \mathbf {x} _{k}\in \mathbb {R} ^{p}}$ and ${\displaystyle \mathbf {y} _{k}\in \mathbb {R} ^{q}}$. The main difference compared to previously is that the matrix ${\displaystyle \mathbf {H} }$ now is ${\displaystyle p\times p}$ and anti-symmetric. When ${\displaystyle p>2}$ the space of such matrices is no longer one-dimensional, it is of dimension

${\displaystyle M={\frac {p\,(p-1)}{2}}.}$

This means that each value of k provides M homogeneous equations of the type

${\displaystyle 0=\mathbf {x} _{k}^{T}\,\mathbf {H} _{m}\,\mathbf {A} \,\mathbf {y} _{k}}$   for   ${\displaystyle \,m=1,\ldots ,M}$   and for ${\displaystyle \,k=1,\ldots ,N}$

where ${\displaystyle \mathbf {H} _{m}}$ is a M-dimensional basis of the space of ${\displaystyle p\times p}$ anti-symmetric matrices.

References

• Richard Hartley and Andrew Zisserman (2003). Multiple View Geometry in computer vision. Cambridge University Press. ISBN 978-0-521-54051-3.

External links

• Homography Estimation by Elan Dubrofsky (§2.1 sketches the "Basic DLT Algorithm")

• A DLT Solver based on MATLAB by Hsiang-Jen (Johnny) Chien